74 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA of V which converges to/e V with respect to the weak topology (i.e., is such that lim fn(x) =f(x) for all x e E) is also said to be weakly convergent. On n~>oo the same principle, we shall speak of subsets of V being weakly closed (in V), weakly compact, etc. A subset B of KE is said to be weakly bounded if all the seminorms (12.15.2) are bounded on B, that is if sup \f(x)\ < +00 for /6B all x e E. We shall also use the phrases weakly continuous (resp. weakly differ entiable, weakly analytic) or scalarly continuous (resp. scalarly differen- tiable, scalarly analytic) in place of " continuous (resp. differentiate, analytic) with respect to the weak topology." If E is a locally convex space (over K = R or C), the vector space of con- tinuous linear forms on E is called the dual of E, and is often denoted by E'. If xf e E' is a continuous linear form on E, its value at a point x e E is often denoted by the expression <x, x'> or <X, xy instead of x'(x). Since E' is a vector subspace of KB, it is endowed with the weak topology defined by the seminorms x't-*\(x, x'y\. For each xeE, the. function x'\—>(x,x'y is a weakly continuous linear form on E'. If E is a Frechet space and E0 a dense vector subspace of E, then it follows from (12.9.4) and the principle of extension of identities (3.15.2) that the mapping x't-tx' |E0 is an (algebraic) isomorphism of the dual E' of E onto the dual EJ of E0 . (12.15.3) Let E, F be two locally convex spaces, E' and F' their respective duals, and u : E -> F a continuous linear mapping. Then y'\-+y' ° u is a linear mapping ofF' into E' which is continuous with respect to the weak topologies. If y' is a continuous linear form on F, then yr ° u is a continuous linear form on E. Given x0 e E, there exists y0 e F such that <%0 , y' o w> = O'os.jO for all yr e F', namely yQ = w(jc0); the continuity of the mapping y'\-»yfou is therefore an immediate consequence of (12.14.11) and the definition (12.15.2) of the seminorms which define the weak topology. We write yr o u = tu(y')9 and the linear mapping lu : F' -> E' is called the transpose of the continuous linear mapping u. Thus we have (12.15.4) <«<*), /> = <*,'« for all x e E and y' e F'. Clearly (1 2.1 5.5) 'fa + u2) = \ + * u2 , \Xu) =A'*u for any scalar 1, and for any continuous linear mappings ui, u2 of E into F. Ifble metrizable subspace of F such that /(E) n L is